I would like to shed some light, even though I'm no light bringer

Firstly, the languages that we use throughout the book so far contain only a's and b's. I think this corresponds nicely with binary in computers. So we are essentially working with binary strings. More specifically, binary strings which will either bring a machine (computer) to a desired or undesired state.

The pumping lemma is used exclusively (I think) to distinguish a regular language or irregular language.

The pumping lemma states (there's a theorem somewhere) that if a language were regular, there would exist 3 strings (or binary combinations) of x, y, z such that xyz and xyyz were both words in this language. (This is verbatim from the bottom of page 193)

Pumping lemmas prove irregular languages (I think) by contradiction. The string is analyzed in the context of the language and the language is then proved regular or irregular.

It is clear from the given string structure that there can exist only one b, slap in the middle of that string. So observation 1 makes sense by intuition.

Observation 2 refers to the lemma. It observes that if the y part of the string (from the abovementioned words allowed in the language by the lemma theorem) were all a's, then the b part of the string (there must be a b, by the definition given of this language) must be either in the x part or the z part.

If this were so *puts on Einstein wig* the string xyyz has increased the number of a's either before or after the b in the string, but not both. This is contrary to the language definition.

*takes off Einstein wig* Then they draw those two conclusions.

Hope this helps (and is correct).